Question: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 5x + 48$, and $ m \angle ABC = 8x - 23$, find $m\angle ABC$. $B$ $A$ $D$ $C$
Solution: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {8x - 23} + {5x + 48} = {90}$ Combine like terms: $ 13x + 25 = 90$ Subtract $25$ from both sides: $ 13x = 65$ Divide both sides by $13$ to find $x$ $ x = 5$ Substitute $5$ for $x$ in the expression that was given for $m\angle ABC$ $ m\angle ABC = 8({5}) - 23$ Simplify: $ {m\angle ABC = 40 - 23}$ So ${m\angle ABC = 17}$.